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Statistical Analysis of Stationary Time Series
Ulf Grenander and Murray Rosenblatt

AMS Chelsea Publishing
1957; 308 pp; hardcover
Volume: 320
ISBN-10: 0-8218-4437-7
ISBN-13: 978-0-8218-4437-3
List Price: US$41
Member Price: US$36.90
Order Code: CHEL/320.H
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From the Preface to the First Edition (1957): "The purpose of this book is two-fold. It is written in the terminology of the theoretical statistician because one of our objectives is to direct his attention to an approach to time series analysis that is essentially different from most of the techniques used by time series analysts in the past. The second objective is to present a unified treatment of methods that are being used increasingly in the physical sciences and technology. We hope that the book will be of considerable interest to research workers in these fields. Keeping the first objective in mind, we have given a rigorous mathematical discussion of these new topics in time series analysis. The existing literature in time series analysis is characterized with few exceptions by a lack of precision both in conception and in the mathematical treatment of the problems dealt with. To avoid this vagueness, we have devoted more space to rigorous proofs than may appear necessary to some readers, but we believe that a study of the proofs will furnish valuable clues to the practical validity of the results and be an important guide to intuition. We have tried to balance the formal proofs with intuitive remarks and comments on practical applications. While the regularity assumptions we have required in many cases may seem restrictive, appropriately interpreted they give an indication of the range in which the methods are practically valid. We have made such interpretations in the comments accompanying the formal proofs."

Readers should have knowledge of statistics and basic probability. The second edition was printed with corrections.


Graduate students and research mathematicians.

Table of Contents

  • Stationary Stochastic Processes and Their Representations: 1.0 Introduction; 1.1 What is a stochastic process?; 1.2 Continuity in the mean; 1.3 Stochastic set functions of orthogonal increments; 1.4 Orthogonal representations of stochastic processes; 1.5 Stationary processes; 1.6 Representations of stationary processes; 1.7 Time and ensemble averages; 1.8 Vector processes; 1.9 Operations on stationary processes; 1.10 Harmonizable stochastic processes
  • Statistical Questions when the Spectrum is Known (Least Squares Theory): 2.0 Introduction; 2.1 Preliminaries; 2.2 Prediction; 2.3 Interpolation; 2.4 Filtering of stationary processes; 2.5 Treatment of linear hypotheses with specified spectrum
  • Statistical Analysis of Parametric Models: 3.0 Introduction; 3.1 Periodogram analysis; 3.2 The variate difference method; 3.3 Effect of smoothing of time series (Slutzky's theorem); 3.4 Serial correlation coefficients for normal white noise; 3.5 Approximate distributions of quadratic forms; 3.6 Testing autoregressive schemes and moving averages; 3.7 Estimation and the asymptotic distribution of the coefficients of an autoregressive scheme; 3.8 Discussion of the methods described in this chapter
  • Estimation of the Spectrum: 4.0 Introduction; 4.1 A general class of estimates; 4.2 An optimum property of spectrograph estimates; 4.3 A remark on the bias of spectrograph estimates; 4.4 The asymptotic variance of spectrograph estimates; 4.5 Another class of estimates; 4.6 Special estimates of the spectral density; 4.7 The mean square error of estimates; 4.8 An example from statistical optics
  • Applications: 5.0 Introduction; 5.1 Derivations of spectra of random noise; 5.2 Measuring noise spectra; 5.3 Turbulence; 5.4 Measuring turbulence spectra; 5.5 Basic ideas in a statistical theory of ocean waves; 5.6 Other applications
  • Distribution of Spectral Estimates: 6.0 Introduction; 6.1 Preliminary remarks; 6.2 A heuristic derivation of a limit theorem; 6.3 Preliminary considerations; 6.4 Treatment of pure white noise; 6.5 The general theorem; 6.6 The normal case; 6.7 Remarks on the nonnormal case; 6.8 Spectral analysis with a regression present; 6.9 Alternative estimates of the spectral distribution function; 6.10 Alternative statistics and the corresponding limit theorems; 6.11 Confidence band for the spectral density; 6.12 Spectral analysis of some artificially generated time series
  • Problems in Linear Estimation: 7.0 Preliminary discussion; 7.1 Estimating regression coefficients; 7.2 The regression spectrum; 7.3 Asymptotic expression for the covariance matrices; 7.4 Elements of the spectrum; 7.5 Polynomial and trigonometric regression; 7.6 More general trigonometric and polynomial regression; 7.7 Some other types of regression; 7.8 Detection of signals in noise; 7.9 Confidence intervals and tests
  • Assorted Problems: 8.0 Introduction; 8.1 Prediction when the conjectured spectrum is not the true one; 8.2 Uniform convergence of the estimated spectral density to the true spectral density; 8.3 The asymptotic distribution of an integral of a spectrograph estimate; 8.4 The mean square error of prediction when the spectrum is estimated; 8.5 Other types of estimates of the spectrum; 8.6 The zeros and maxima of stationary stochastic processes; 8.7 Prefiltering of a time series; 8.8 Comments on tests of normality
  • Problems
  • Appendix on complex variable theory
  • Bibliography
  • Index
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